Sir William Rowan Hamiuton on Equations of the Fifth Degree. 363 



T,,,, = (- 2 - «.' + ^-^ + 2u>f, (151) 



and 



T,3,, + Tj,,,, = 0. (152) 



Again, 



T3^ = (1 _ a^-^r (2 - «)^ T,,^ = (1 - u,-f (2 - 0,^ ; (153) 



and if we make 



(2 — «.)^ = E-o, (2 +«,)*= E + o, (154) 



we shall have 



Ezi 32 + 80w^+10wS o = 80w + 40«)^ -f o;^ ; (155) 



also, 



(1 — uPf = - 5«)^(1 - w^) (1 - w- + w*) ; (156) 



we find, therefore, by easy calculations, 



(1 — wy E = 300 4- 430« - llOw^ — 540«.' — SOw\ ] 

 (1 — w^)^ = 600 +190« — 405«.^ — 395«)^ + 10«)''; j 



and by subtracting the latter of these two products from the former, and after- 

 wards changing w to its reciprocal, we obtain : 



T3254 = - 300 + 240co + 295«»^ - 1 45«»-' — 90u,\ ] ^'^^ 



T^= — 300 + 2400."+ 295w^ — 145tt.-^ — 90«. j 



We have, therefore, by (20), 



T32M + T«.3==-750; (159) 



and, consequently, by (33) and (152), 



v,.= -^5. (160) 



34. In like manner, to compute, in this example, the second of the six func- ' 

 tions V, we have 



adding then the two products (157) together, and afterwards changing w to w^ 

 and w^ successively, we find, by (154) : 



3 a2 



